Finite Group Databasebeta

← #37574538

Subgroups — #37574538

Order 24448.

248 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 41 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27133405C211yesmaximal, characteristic, complemented
2#27132838C212yesmaximal, complemented
3#27132835C211yesmaximal, characteristic, complemented
4#27132459C212yesmaximal, complemented
5#355×C191C211yesmaximal, characteristic, complemented
6#20259370C2211yescharacteristic, complemented
7#20259350—22no—
8#20259348—21no—
9#20259311C2212yes—
10#20259310—21no—
11#20259308—22no—
12#20259292—22no—
13#20259290—21no—
14#20259250—21no—
15#20259248—22no—
16#20259241C2211yescharacteristic, complemented
17#124×C191C2211yescharacteristic
18#101×C191C2212yescomplemented
19C23×D191—21no—
20#15690288—41no—
21#15690287—21no—
22#15690287—22no—
23C2×Dic382—21no—
24Q8×D191—41no—
25C2×D764—21no—
26C2×D764D412yes—
27C2×C22⋊D191—22no—
28#15690276—42no—
29#15690275—41no—
30D1528—42no—
31#15690272—41no—
32#15690271—21no—
33C22⋊Dic191—21no—
34(C2×D191)⋊C4—22no—
35(C2×D191)⋊C4—22no—
36(C2×D191)⋊C4—22no—
37C4×Dic191D412yes—
38C2×Q8×C191C2311yescharacteristic
39C2×D4×C191D411yescharacteristic
40M16×C191—22no—
41C22⋊C4×C191—22no—
42C42×C191D411yescharacteristic
43C22×D191—22no—
44C22×D191—82no—
45C22×D191—81no—
46Dic382—21no—
47Dic382—22no—
48Dic382—81no—
49D764—21no—
50D764—22no—
51D764—42no—
52D764—81no—
53C22⋊D191—41no—
54C22⋊D191—82no—
55C2×Dic191—22no—
56C4×D191—42no—
57C4×D191—41no—
58C191⋊C8—41no—
59C23×C191—21no—
60Q8×C191—41no—
61D4×C191—41no—
62C2×C4×C191—21no—
63C2×C4×C191C2×D412yes—
64C2×C4×C191C2×D411yescharacteristic
65C8×C191—42no—
66C22×C191—42no—
67C22×C191—41no—
68C22×C191#12011yescharacteristic
69C2×D191—42no—
70C2×D191—42no—
71C2×D191—41no—
72C2×D191—42no—
73C2×D191—84no—
74C4×C191—22no—
75C4×C191—21no—
76C4×C191—41no—
77Dic191—42no—
78Dic191—41no—
79C2×C191—21no—
80C2×C191—81no—
81C2×C191B(4,𝔽2)11yescharacteristic
82D191—81no—
83D191—82no—
84C191#220211yescharacteristic, complemented, Sylow
85#2202—1911nomaximal, Sylow
86H(3,𝔽4)—1911no—
87#454—1911no—
88#454—1912no—
89#355—1912no—
90#355D19111yescharacteristic, complemented
912−1+4—1911no—
92AGL(1,ℤ/8)—3822no—
93AGL(1,ℤ/8)—3821no—
94#124—3822no—
95#124—3821no—
96#124—1912no—
97#124C2×D19111yescharacteristic
98#120—3822no—
99#120—3821no—
100#104—3822no—
101#104—3821no—
102#101—1911no—
103#101C2×D19112yescomplemented
104C24—3821no—
105#41—3822no—
106#41—3821no—
107#41—7641no—
108C2×Q8—3821no—
109C2×Q8—7641no—
110C2×Q8C22×D19111yescharacteristic
111C2×D4—3821no—
112C2×D4—3822no—
113C2×D4—1912no—
114C2×D4D76411yescharacteristic
115SD16—7642no—
116SD16—7641no—
117D8—7642no—
118D8—7641no—
119M16—22no—
120M16—3821no—
121C22⋊C4—22no—
122C22⋊C4—3822no—
123C22⋊C4—3821no—
124C22⋊C4—3822no—
125C22⋊C4—3822no—
126C42—1912no—
127C42D76411yescharacteristic
128C23—21no—
129C23—3822no—
130C23—15282no—
131C23—15281no—
132Q8—41no—
133Q8—3822no—
134Q8—3821no—
135Q8—15281no—
136D4—41no—
137D4—3822no—
138D4—3821no—
139D4—15281no—
140D4—15282no—
141D4—7642no—
142D4—7641no—
143C2×C4—21no—
144C2×C4—3822no—
145C2×C4—7642no—
146C2×C4—7641no—
147C2×C4D4×D19112yes—
148C2×C4C2×D76411yescharacteristic
149C8—42no—
150C8—7641no—
151C22—42no—
152C22—41no—
153C22—15284no—
154C22—7642no—
155C22—7642no—
156C22—7641no—
157C22—7642no—
158C22#2025929211yescharacteristic
159C4—22no—
160C4—21no—
161C4—41no—
162C4—7642no—
163C4—7641no—
164C2—21no—
165C2—81no—
166C2—15282no—
167C2—15281no—
168C2#2713285111yescharacteristic
subgroupG/H
centerC2#27132851
commutator, cosocleC2×Q8×C191C23
FrattiniC2×Q8C22×D191
Fitting#355×C191C2
socleC2×C191B(4,𝔽2)
hypercenter#355D191
hypocenterC191#2202
LCSG ▷ C2×Q8×C191 ▷ C22×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #27133405C2semidirect
2. #27132838C2semidirect
3. #27132835C2semidirect
4. #27132459C2semidirect
5. #355×C191C2semidirect
6. #20259370C22semidirect
16. #20259241C22semidirect
18. #101×C191C22semidirect
84. C191#2202semidirect
90. #355D191semidirect
103. #101C2×D191semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#271334052
#271328382
#271328352
#271324592
#355×C1912
#2202191